The quantitative convergence analysis of Sinkhorn is quantified using two norms.
The first one is the sup-norm \({_\infty \max |, and the second one is the Hilbert pseudo-norm {_\star \inf _{\in }{ }_\infty = \tfrac{1}{2}(\max  \min , which compares functions up to constant translations.The Hilbert norm is useful to analyze the convergence of balanced Sinkhorn algorithm, as well as TI-Sinkhorn, since iterates share the same invariance to translations.By contrast, the convergence of unbalanced Sinkorn is quantified for the sup-norm since it is not invariant.}}\) To make the comparison between iterates of Sinkhorn and TI-Sinkhorn informative, one first computes iterates \((_t,_t)\)  with \((\Psi _1,\Psi _2)\) , then compares their translation \(\Phi (_t,_t)\)  to the optimal potentials \((\star ,^\star )\)  of \(_\epsilon \) .
By definition of \(_\epsilon \)  and \(\Phi \)  (Definition REF  and Equation (REF )), if \((_t,_t)\)  are optimal for \(_\) , then \(\Phi (_t,_t)\)  are optimal for \(_\epsilon \) .
It yields the following convergence rates when \({\phi _i}=\rho _i\) . It combines results from [1]}, [2]}.
